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log 15 ( a ) = b

15 b = a

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log 13 ( 142 ) = a

13 a = 142

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For the following exercises, rewrite each equation in logarithmic form.

n 4 = 103

log n ( 103 ) = 4

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y x = 39 100

log y ( 39 100 ) = x

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For the following exercises, solve for x by converting the logarithmic equation to exponential form.

log 2 ( x ) = 3

x = 2 3 = 1 8

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log 3 ( x ) = 3

x = 3 3 = 27

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log 9 ( x ) = 1 2

x = 9 1 2 = 3

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log 6 ( x ) = 3

x = 6 3 = 1 216

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For the following exercises, use the definition of common and natural logarithms to simplify.

e ln ( 10.125 ) + 4

14.125

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Numeric

For the following exercises, evaluate the base b logarithmic expression without using a calculator.

For the following exercises, evaluate the common logarithmic expression without using a calculator.

2 log ( 100 3 )

12

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For the following exercises, evaluate the natural logarithmic expression without using a calculator.

ln ( e 0.225 ) 3

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Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

Extensions

Is x = 0 in the domain of the function f ( x ) = log ( x ) ? If so, what is the value of the function when x = 0 ? Verify the result.

No, the function has no defined value for x = 0. To verify, suppose x = 0 is in the domain of the function f ( x ) = log ( x ) . Then there is some number n such that n = log ( 0 ) . Rewriting as an exponential equation gives: 10 n = 0 , which is impossible since no such real number n exists. Therefore, x = 0 is not the domain of the function f ( x ) = log ( x ) .

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Is f ( x ) = 0 in the range of the function f ( x ) = log ( x ) ? If so, for what value of x ? Verify the result.

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Is there a number x such that ln x = 2 ? If so, what is that number? Verify the result.

Yes. Suppose there exists a real number x such that ln x = 2. Rewriting as an exponential equation gives x = e 2 , which is a real number. To verify, let x = e 2 . Then, by definition, ln ( x ) = ln ( e 2 ) = 2.

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Is the following true: log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? Verify the result.

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Is the following true: ln ( e 1.725 ) ln ( 1 ) = 1.725 ? Verify the result.

No; ln ( 1 ) = 0 , so ln ( e 1.725 ) ln ( 1 ) is undefined.

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Real-world applications

The exposure index E I for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation E I = log 2 ( f 2 t ) , where f is the “f-stop” setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

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Refer to the previous exercise. Suppose the light meter on a camera indicates an E I of 2 , and the desired exposure time is 16 seconds. What should the f-stop setting be?

2

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The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log I 1 I 2 = M 1 M 2 where M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

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Questions & Answers

what is mutation
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classification of plants, gymnosperm features.
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State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
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malaria is caused by an insect called mosquito.
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Malaria is cause by female anopheles mosquito
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Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
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what are pathogens
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In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
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A virus
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Definition of respiration
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respiration is the process in which we breath in oxygen and breath out carbon dioxide
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in the mouth
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stimulates the follicle to release the mature ovum into the oviduct
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endocrine secrete hormone and regulate body process
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Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
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Biology is the study of how living organisms live and survive in a specific environment
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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